- Identify the hypotenuse \(AB\) as the diameter of the circle because the triangle is right-angled at \(C\).
- Calculate the center of the circle as the midpoint of the diameter \(AB\), resulting in \((-1, 2)\).
- Determine the squared radius \(r^2\) as one-fourth of the square of the distance between \(A\) and \(B\), which equals \(8\).
- Write the final equation of the circle in standard form: \(\boxed{(x + 1)^2 + (y - 2)^2 = 8}\).
Explanation
- Analyze the geometry of the problem.
The points given are \(A(-3, 0)\), \(B(1, 4)\), and \(C(-3, 4)\). Since triangle \(ABC\) is a right-angled triangle with the right angle at \(C\), the side \(AB\) is the hypotenuse. According to Thales' Theorem, the hypotenuse of a right-angled triangle inscribed in a circle is a diameter of that circle. Thus, the segment \(AB\) is the diameter of the required circle.
- Calculate the center of the circle.
The diameter endpoints are \(A(-3, 0)\) and \(B(1, 4)\). The center \((h, k)\) of the circle is the midpoint of the diameter \(AB\):
\[h = \frac{x_A + x_B}{2} = \frac{-3 + 1}{2} = \frac{-2}{2} = -1\]
\[k = \frac{y_A + y_B}{2} = \frac{0 + 4}{2} = \frac{4}{2} = 2\]
So, the center of the circle is \((-1, 2)\).
- Calculate the squared radius of the circle.
The radius \(r\) is half the distance between \(A(-3, 0)\) and \(B(1, 4)\). First, we find the squared length of the diameter \(d^2 = (x_B - x_A)^2 + (y_B - y_A)^2\):
\[d^2 = (1 - (-3))^2 + (4 - 0)^2 = 4^2 + 4^2 = 16 + 16 = 32\]
The radius squared is \(r^2 = \left( \frac{d}{2} \right)^2 = \frac{d^2}{4} = \frac{32}{4} = 8\).
- Formulate the final equation of the circle.
Using the standard form of the circle equation, \((x - h)^2 + (y - k)^2 = r^2\), we substitute the center \((h, k) = (-1, 2)\) and \(r^2 = 8\):
\[(x - (-1))^2 + (y - 2)^2 = 8\]
\[(x + 1)^2 + (y - 2)^2 = 8\]
This is the equation of the circle passing through points \(A\), \(B\), and \(C\). The final answer is \(\boxed{(x + 1)^2 + (y - 2)^2 = 8}\).
Examples
In engineering and architecture, finding the circle passing through specific points is essential for designing circular arches or support structures. For instance, if you are designing a bridge with a curved span and you know three specific contact points on the ground or existing pillars, calculating the equation of the circle helps determine the precise curvature required for the structural arch, ensuring it is balanced and structurally sound. This same mathematical principle of using a right-angled triangle to define a diameter is often applied in surveying and site planning to set out perfect circular layouts for foundations.